Questions: In Desmos, draw the graph of the first derivative function and interpret it in the context of your application problem. P(t) = t^2 / (0.4 t^2 + 1)

Solution Steps

Given the function: \[ P(t) = \frac{t^2}{0.4t^2 + 1} \]

We use the quotient rule to find the first derivative: \[ P'(t) = \frac{(0.4t^2 + 1) \cdot 2t - t^2 \cdot (0.8t)}{(0.4t^2 + 1)^2} \]

Simplify the numerator: \[ P'(t) = \frac{2t(0.4t^2 + 1) - 0.8t^3}{(0.4t^2 + 1)^2} \] \[ P'(t) = \frac{0.8t^3 + 2t - 0.8t^3}{(0.4t^2 + 1)^2} \] \[ P'(t) = \frac{2t}{(0.4t^2 + 1)^2} \]

Final Answer